A theoretical calculation of a smoothly bent Haldane lattice model has found a distinct change in its third-order transverse energy response as the model crosses between topological regimes. The extrapolated size of that discontinuity was about 0.00672 in the reported units, close to a relativistic benchmark of about 0.00663. The absolute response itself was not universal: it retained lattice-dependent contributions.
The analyzed data were calculated response functions for a Haldane model on smoothly curved graphene. The calculation follows a smooth out-of-plane bend that varies along x while preserving translational invariance along y. It treats changes in nearest-neighbor hoppings and an effective scalar potential as separate pieces of the deformation.
Charge and energy split apart
The split between those two pieces is central. At leading gradient order, only the scalar-potential term contributed to the transverse charge current. The hopping-modulation term made no contribution at that order.
For transverse energy current, the pattern was reversed within the orders examined. The scalar potential produced no response, while the reported current was attributed entirely to nearest-neighbor hopping modulation. This result is limited to the gradient orders included in the calculation.
The topological marker
The relevant marker was the Chern number of the occupied band. It was zero in the trivial regime and quantized at plus or minus one in the topological regimes, with the sign set by the model parameter t2.
Not every part of the energy response carried that same marker. The leading first-order term was nonuniversal and depended on how individual bonds were deformed. When all three nearest-neighbor hoppings were modulated symmetrically, that term vanished. The calculation therefore separates a bond-dependent leading term from the third-order transition discontinuity.
How the model was built
To model the bend, the authors used a Slater-Koster parameterization for curvature-modified nearest-neighbor hopping. The pure-bending setup also neglects changes in nearest-neighbor bond lengths. The scalar potential was included phenomenologically rather than fixed by a microscopic calculation.
For the energy calculation, the energy-current operator was derived from the lattice continuity equation with an auxiliary gravitational field coupled to local energy density. The response was expanded term by term through third order in momentum transfer along x. After Brillouin-zone integration, equilibrium and contact terms vanished, leaving only the calculation's bubble contribution.
A result tied to its assumptions
The orientation check added another layer to the result. The bond-summed third-order response was independent of crystal orientation, although the real-space third-order current showed weak orientation dependence. Changing the crystal's angle therefore did not alter the summed third-order signal in the calculation, even though local current amplitudes changed slightly.
The findings remain bounded by the model. The scalar potential is phenomenological, the geometry assumes pure bending, and the absolute third-order response contains nonuniversal lattice contributions away from the transition. The reported discontinuity is an extrapolated value, so the calculation does not establish a universal energy-response magnitude away from the transition.
The supplied document is an arXiv preprint, version 2. Its evidence consists of analytical and numerical response functions for the curved Haldane model, so the reported discontinuity is a result of that model calculation.
Paper data and sources
Original title: Topological signatures in the curvature-induced energy response
Authors: Jaehyeok Lee, Iuegyun Hong, Jinhong Park
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text